Arcane Disciple when some spells are already on spell list

Was just looking for a bit of insight into the Arcane Disciple feat. A quick quote on what that does:

Add the chosen domain’s spells to your class list of arcane spells. If you have arcane spellcasting ability from more than one class, you must pick which arcane spellcasting ability this feat applies to. Once chosen, this decision cannot be changed for that feat. You may learn these spells as normal for your class; however, you use Wisdom (rather than the normal ability for your spellcasting) when determining the save DC for the spell. In addition, you must have a Wisdom score equal to 10 + the spell’s level in order to prepare or cast a spell gained from this feat. Each day, you may prepare (or cast, if you cast spells without preparation) a maximum of one of these domains spells of each level.

I’m looking into a summoner build, and want to get into Thaumaturgist as a sorcerer. I figured I’d take Arcane Disciple and pick the summoning domain for Lesser Planar Ally. As a sorcerer though, all the Summon Monster spells and Gate are already Sor/Wiz spells. Does this mean I now need 19 WIS to cast Gate and the appropriate WIS for Summon Monster 1/2/3/5/7? Or can I still use these spells as a sorcerer and just rely on CHA for the main stat?

For further context, Vow of Poverty is part of the character build for a bit of game-balancing flavour. Just wondering if I need to allocate one of my 4 stat increases to WIS. Otherwise I’d rather put my points into just about anything else. Has anyone seen any rules regarding this? Or is it just a straight "the spells in this domain list now rely on WIS", regardless of whether or not they were already on your class list?